K-Theory of curves over number fields

We consider the algebraic K-groups with coefficients of smooth curves over number fields. We give a proof of the Quillen-Lichtenbaum conjecture at the prime 2 and prove explicit corank formulas for the algebraic K-groups with divisible coefficients. At odd primes these formulas assume the Bloch-Kato conjecture, at the prime 2 the formulas hold nonconjecturally.     

The p-adic Gross-Zagier formula for elliptic curves at supersingular primes

Let p be a prime number and let E be an elliptic curve defined over ? of conductor N. Let K be an imaginary quadratic field with discriminant prime to pN such that all prime factors of N split in K. B. Perrin-Riou established the p-adic Gross-Zagier formula that relates the first derivative of the p-adic L-function of E over K to the p-adic height of the Heegner point for K when E has good ordinary reduction at p. In this article, we prove the p-adic Gross-Zagier formula of E for the cyclotomic ? _p -extension at good supersingular prime p. Our result has an application for the full Birch and Swinnerton-Dyer conjecture. Suppose that the analytic rank of E over ? is 1 and assume that the Iwasawa main conjecture is true for all good primes and the p-adic height pairing is not identically equal to zero for all good ordinary primes, then our result implies the full Birch and Swinnerton-Dyer conjecture up to bad primes. In particular, if E has complex multiplication and of analytic rank 1, the full Birch and Swinnerton-Dyer conjecture is true up to a power of bad primes and 2.     

The prime at infinity and the rank of the class group in global function fields

In this paper we construct, for any integers m and n, and 2?g?m-1, infinitely many function fields K of degree m over F(T) such that the prime at infinity splits into exactly g primes in K and the ideal class group of K contains a subgroup isomorphic to (Z/nZ)^m-g. This extends previous results of the author and Lee for the cases g=1 and g=m.     

Hasse invariants and anomalous primes for elliptic curves with complex multiplication

Let C be an elliptic curve defined over Q. Let p be a prime where C has good reduction. By definition, p is anomalous for C if the Hasse invariant at p is congruent to 1 modulo p. The phenomenon of anomalous primes has been shown by Mazur to be of great interest in the study of rational points in towers of number fields. This paper is devoted to discussing the Hasse invariants and the anomalous primes of elliptic curves admitting complex multiplication. The two special cases Y² = X³ + a₄X and Y² = X³ + a₆ are studied at considerable length. As corollaries, some results in elementary number theory concerning the residue classes of the binomial coefficients (_n²ⁿ) (Resp. (_n³ⁿ)) modulo a prime p = 4n + 1 (resp. p = 6n + 1) are obtained. It is shown that certain classes of elliptic curves admitting complex multiplication do not have any anomalous primes and that others admit only very few anomalous primes.     

Bounds on the number of automorphisms of curves over algebraically closed fields

In this note we compare the maximum number of automorphisms of smooth projective curves of genus g ≥ 2 over algebraically closed fields of characteristic 0 or a prime p > 0, respectively. For given g ≥ 2, we show that for almost all primes p > 0, this number is the same.     

Asymptotic behavior of number fields with prescribed ℓ-class numbers

Let K be a cyclic Galois extension of the rational numbers Q of degree ?, where ? is a prime number. Let h_? denote the order of the Sylow ?-subgroup of the ideal class group of K. If h_? = ?^s(s ≥ 0), it is known that the number of (finite) primes that ramify in K/Q is at most s + 1 (or s + 2 if K is real quadratic). This paper shows that “most” of these fields K with h_? = ?^s have exactly s + 1 ramified primes (or s + 2 ramified primes if K is real quadratic). Furthermore the Sylow ?-subgroup of the ideal class group is elementary abelian when h_? = ?^s and there are s + 1 ramified primes (or s + 2 ramified primes if K is real quadratic).     

Weak Leopoldt's conjecture for Hecke characters of imaginary quadratic fields

We give a proof of the weak Leopoldt's conjecture à la Perrin-Riou, under some technical condition, for the p-adic realizations of the motive associated to Hecke characters over an imaginary quadratic field K of class number 1, where p is a prime >3 and where the CM elliptic curve associated to the Hecke character has good reduction at the primes above p in K. This proof makes use of the 2-variable Iwasawa main conjecture proved by Rubin. Thus we prove the Jannsen conjecture for the above p-adic realizations for almost all Tate twists.     

Stability theory for absolutely continuous parts of commutative m-tuples of self-adjoint operators in complex Hilbert spaces

By the method of direction wave operators, we prove that absolutely continuous parts of commutative m-tuples of self-adjoint operators in complex Hilbert spaces are invariant up to unitary equivalence under perturbations of class B_pfor p < m, m ? 2.     

Infinite class field towers

This paper studies infinite class field towers of number fields K that are ramified over only at one finite prime. In particular, we show the existence of such towers for a general family of primes including p = 2, 3 and 5.     

Finite prime-field characteristic sets for planar configurations

Examples are constructed of planar matroids with finite prime-field characteristic sets (i.e. matroids representable over a finite set of prime fields but over fields of no other characteristic). In particular, for any n>3, a projectively unique integer matrix is constructed with 2lsqblog₂nrsqb+6 columns which often gives nonsingleton characteristic sets and, when n is prime, has characteristic set {n}. Many finite subsets of primes are shown to be characteristic sets, including {23,59} (the smallest pair found using these methods), all pairs of primes {p, p′:67?p<p′?293}, and the seventeen largest five-digit primes. Probabilistic arguments are presented to support the conjecture that prime-field characteristic sets exist of every finite cardinality. For p>3, AG(2,p) is shown to be a subset of PG(2, q) only for q=p^s. Another general construction technique suggests that when $\text{P}\text{={p}{}_1\text{,…,p}{}_{\mathrm{k}}\text{}}$ are the primitive prime divisors of 2ⁿ±1 (n sufficiently large), then there is a matroid with O (log n) points whose characteristic set is $\text{P}$. We remark that although only one finite nonsingleton characteristic set (due to R. Reid) was known prior to this paper, a new technique by J. Kahn has shown that every finite set of primes forms a (non-prime-field) characteristic set.     

Codes from lattice and related graphs, and permutation decoding

Codes of length n² and dimension 2n−1 or 2n−2 over the field F_p, for any prime p, that can be obtained from designs associated with the complete bipartite graph K_n,n and its line graph, the lattice graph, are examined. The parameters of the codes for all primes are obtained and PD-sets are found for full permutation decoding for all integers n≥3.     

Primitive roots in quadratic fields, II

We consider an analogue of Artin's primitive root conjecture for algebraic numbers which are not units in quadratic fields. Given such an algebraic number α, for a rational prime p which is inert in the field, the maximal possible order of α modulo (p) is p²−1. An extension of Artin's conjecture is that there are infinitely many such inert primes for which this order is maximal. We show that for any choice of 113 algebraic numbers satisfying a certain simple restriction, at least one of the algebraic numbers has order at least for infinitely many inert primes p.     

On the cyclotomic polynomial

For a given positive integer m and an algebraic number field K necessary and sufficient conditions for the mth cyclotomic polynomial to have K-integral solutions modulo a given integer of K are given. Among applications thereof are: that the solvability of the cyclotomic polynomial mod an integer yields information about the class number of related number fields; and about representation of integers by binary quadratic forms. The latter extends previous work of the author. Moreover some information is obtained pertaining to when an integer of K is the norm of an integer in a given quadratic extension of K. Finally an explicit determination of the pqth cyclotomic polynomial for distinct primes p and q is provided, and known results in the literature as well as generalizations thereof are obtained.     

On the vanishing of Selmer groups for elliptic curves over ring class fields

Let E/Q be an elliptic curve of conductor N without complex multiplication and let K be an imaginary quadratic field of discriminant D prime to N. Assume that the number of primes dividing N and inert in K is odd, and let Hc be the ring class field of K of conductor c prime to ND with Galois group Gc over K. Fix a complex character χ of Gc. Our main result is that if LK(E,χ,1)≠0 then Selp(E/Hc)χ_⊗W=0 for all but finitely many primes p, where Selp(E/Hc) is the p-Selmer group of E over Hc and W is a suitable finite extension of Zp containing the values of χ. Our work extends results of Bertolini and Darmon to almost all non-ordinary primes p and also offers alternative proofs of a χ-twisted version of the Birch and Swinnerton-Dyer conjecture for E over Hc (Bertolini and Darmon) and of the vanishing of Selp(E/K) for almost all p (Kolyvagin) in the case of analytic rank zero.     

On twin primes associated with the Hawkins random sieve

We establish an asymptotic formula for the number of k-difference twin primes associated with the Hawkins random sieve, which is a probabilistic model of the Eratosthenes sieve. The formula for k=1 was obtained by M.C. Wunderlich [A probabilistic setting for prime number theory, Acta Arith. 26 (1974) 59-81]. We here extend this to k?2 and generalize it to all l-tuples of Hawkins primes.     

The period-index problem in WC-groups I: elliptic curves

Let E/K be an elliptic curve defined over a number field, and let p be a prime number such that E(K) has full p-torsion. We show that the order of the p-part of the Shafarevich-Tate group of E/L is unbounded as L varies over degree p extensions of K. The proof uses O’Neil's period-index obstruction. We deduce the result from the fact that, under the same hypotheses, there exist infinitely many elements of the Weil-Châtelet group of E/K of period p and index p².     

Parity of ranks for elliptic curves with a cyclic isogeny

Let E be an elliptic curve over a number field K which admits a cyclic p-isogeny with p?3 and semistable at primes above p. We determine the root number and the parity of the p-Selmer rank for E/K, in particular confirming the parity conjecture for such curves. We prove the analogous results for p=2 under the additional assumption that E is not supersingular at primes above 2.     

Wild tame-by-cyclic extensions

Suppose G is a semi-direct product of the form Z/pⁿ?Z/m where p is prime and m is relatively prime to p. Suppose K is a complete discrete valuation field of characteristic p>0 with algebraically closed residue field. The main result states necessary and sufficient conditions on the ramification filtrations that occur for wildly ramified G-Galois extensions of K. In addition, we prove that there exists a parameter space for G-Galois extensions of K with given ramification filtration, and we calculate its dimension in terms of the ramification filtration. We provide explicit equations for wild cyclic extensions of K of degree p³.